Last Updated on May 5, 2023

GMAT OFFICIAL GUIDE PS

Solution:

Let’s use the answer choices to help us solve this problem. We are looking for the minimum number of letters that can be used. The smallest number from the answer choices is 4, so let’s ask ourselves this question: Can we use only 4 letters to represent the 12 participants? Assume that the 4 letters are A, B, C and D (keep in mind that for each participant we can use either one letter OR two letters to represent him or her; however if we use two letters, they must be in alphabetical order):

1) A
2) B
3) C
4) D
5) AB
6) AC
7) AD
8) BC
9) BD
10) CD

Under this scheme, we can represent only 10 of the 12 participants. So let’s add in one more letter, say E, and see if having an additional letter allows us to have a unique identifier for each of the 12 participants:

1) A
2) B
3) C
4) D
5) AB
6) AC
7) AD
8) BC
9) BD
10) CD
11) E
12) AE

As you can see, once we’ve added in the letter E we can represent all 12 participants. Since we’ve used A, B, C, D and E, the minimum number of letters that can be used is 5.

Answer: B

Alternate solution:

We can solve this problem by using combinations. We can let n denote the total number of letters needed. Since there are 12 participants and we have the option of one-letter codes and two-letter codes, we need the sum of nC1 and nC2 to be at least 12. That is,

nC1 + nC2 ≥ 12

From here the easiest way to solve this problem is to test the answer choices. We can start with answer choice A.

A. 4

4C1 + 4C2

4 + (4×3)/2

4 + 2×3

10

If we use 4 letters, we will only obtain ten unique identifiers; this is insufficient for 12 participants. Now let’s try answer choice B:

B. 5

5C1 + 5C2

5 + (5×4)/2

5 + 5×2

15

We obtain 15 unique identifier codes by using 5 letters, and this is more than sufficient.

Answer: B